From Symplectic Measurements to the Mahler Conjecture

Abstract

In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains, and Mahler's conjecture on the volume product of centrally symmetric convex bodies. More precisely, we show that if for convex bodies of fixed volume in the classical phase space the Hofer-Zehnder capacity is maximized by the Euclidean ball, then a hypercube is a minimizer for the volume product among centrally symmetric convex bodies.

17 pages, revised version. A generalization of Theorem 1.7 has been added (see Remark 4.2)

References in corpus (2)

• A remark on the Mahler conjecture: local minimality of the unit cube
• The isotropic position and the reverse Santaló inequality

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